statdistribs.Rmd

## Working with statistical distributions

For each statistical distribution, we have function to compute

* density
* distribution function
* quantile function 
* random generation

For the normale distribution `norm`, these are respectively

* `dnorm`
* `pnorm`
* `qnorm`
* `rnorm`

Let's start by sampling 10000 values from a normal distribution $N(0, 1)$:

```{r}
xn <- rnorm(1e6)
hist(xn, freq = FALSE)
rug(xn)
lines(density(xn), lwd = 2)
```

By definition, the area under the density curve is 1. The area at the
left of 0, 1, and 2 are respectively:

```{r}
pnorm(0)
pnorm(1)
pnorm(2)
```

To ask the inverse question, we use the quantile function. The obtain
0.5, `r pnorm(1)` and `r pnorm(2)` of our distribution, we need means
of:

```{r}
qnorm(0.5)
qnorm(pnorm(1))
qnorm(pnorm(2))
```

Finally, the density function gives us the *height* at which we are
for a given mean:

```{r}
hist(xn, freq = FALSE)
lines(density(xn), lwd = 2)
points(0, dnorm(0), pch = 19, col = "red")
points(1, dnorm(1), pch = 19, col = "red")
points(2, dnorm(2), pch = 19, col = "red")
```